Order the following integers from least to greatest:
−41,−51,−38, 50, 41 and 13

Answer:

42 feet

Step-by-step explanation:

This is a ratio question. The ratio of the height of the lamp to its shadow is the same as the ratio of the height of the building to its shadow.

So, 15/9 = 70/x, where x is the building's shadow.

Cross mutiply and solve for x.

15x = 630

x=42

Using the concept of similar triangles, we set up a proportion comparing the heights and shadow lengths of the lamp and building. After solving the proportion 15/9 = 70/x, we find that the shadow cast by the 70-foot tall building is 42 feet long.

To find the length of the shadow cast by the building, we can use the concept of similar triangles. The lamp and its shadow form one triangle, and the building and its shadow form a second triangle. These two triangles are similar because the angles are the same, meaning they have the same shape but are of different sizes.

Given that a 15-foot lamp casts a 9-ft shadow, we can set up the following proportion:
Lamp Height / Lamp Shadow = Building Height / Building Shadow, which simplifies to 15/9 = 70/x, where x is the length of the building's shadow we are trying to find.

By cross-multiplying, we get 15x = 9 * 70, which simplifies to 15x = 630. Dividing both sides by 15 gives us x = 42, so the shadow cast by the building is 42 feet long.

y- -8=(5/2)(x-4)

so is y+8=(5/2)(x-4)

The answer is 11 and 6.

Hope this helps!

The two numbers which sum up to 17 and have a product of 66 are 6 and 11. This was found by using algebra to set up and solve two simultaneous equations.

The question can be resolved using a little bit of algebra. Let's assign the values x and y to these two numbers. We know two things: x + y = 17 (because their sum is 17) and xy = 66 (because their product is 66).

First, you will make y the subject of the first equation making it y = 17 - x. Replace y in the second equation with 17 - x so we will now have x(17 - x) = 66 or 17x - x^2 = 66. Rearranging the equation gives x^2 - 17x + 66 = 0. This equation can be factored to solve for x giving (x - 11)(x - 6) = 0. Therefore, x could be base on the equation 11 or 6.

If x is 11, y will be 6 (because 17 - 11 = 6) and if x is 6, y is 11 (17 - 6 = 11). Therefore, the two numbers are 6 and 11.

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Answer:

pift^2(or your third option) is the area of a circle with a radius of 1.

- The perimeter for it is 8+V10 Units.

The perimeter of the triangle LMN is 8 + √10 units.

Perimeter of a straight sided figures or objects is the total length of it's boundary.

Given is a triangle LMN in the coordinate plane.

The coordinates of the vertices are L(2, 4), M(-2, 1) and N(-1, 4).

We have to find the length of each sides.

Using the distance formula,

LM = [tex]\sqrt{(-2-2)^2+(1-4)^2}[/tex] = √(16 + 9) = √25 = 5

MN = [tex]\sqrt{(-1--2)^2+(4-1)^2}[/tex] = √10

LN = [tex]\sqrt{(-1-2)^2+(4-4)^2}[/tex] = √9 = 3

Perimeter = LM + MN + LN = 8 + √10

Hence the perimeter is 8 + √10 units.

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Your question is incomplete. The complete question is as given below.

Answer:

B

Step-by-step explanation:

Picture shows the formula to find the circumference of a circle

The circumference of a circle is calculated by multiplying the diameter by the mathematical constant π, typically approximated as 3.14159. This relationship is captured by the formula C = πd, which is fundamental in the study of geometry.

The relationship between the circumference and the diameter of a circle is described by the mathematical constant π (pi). The circumference (C) of a circle can be calculated by multiplying the diameter (d) of the circle by π. Therefore, the correct equation is b. Pi times the diameter equals the circumference, which can be expressed as C = πd.

This relationship is a fundamental aspect of Euclidean geometry. It is interesting to note that the diameter of the circle is twice the radius (d = 2r), so the circumference can also be calculated by the formula C = 2πr, where r is the radius. This equation highlights that the circumference is proportional to the diameter, with π serving as the constant of proportionality.

Answer:

B) As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞.

Step-by-step explanation:

Answer:

f(x) = x² - 2x - 15 passes through (-3 , 0) , (0 , -15) , (5 , 0)

f(x) = -x² - 2x + 15 passes through (-2 , 15) , (-1 , 16) , (0 , 15)

f(x) = -x² + 2x - 15 passes through (0 , -15) , (1 , -14) , (2 , -15)

Step-by-step explanation:

* Lets explain how to solve this question

- To find the points whose graph passes through them substitute the

  x-coordinate in the function if the answer is the same with the

  y-coordinate of the point then the graph passes through this point

  lets do that

- Check the first set of points with the first function

# Pint (0 , -15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (0)² - 2(0) - 15 = -15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , -15)

# Pint (1 , -14)

∵ f(x) = x² - 2x - 15

∴ f(0) = (1)² - 2(1) - 15 = -16 ⇒ not same value of y-coordinate

∴ The graph of the function does not pass through point (1 , -14)

∴ The graph does not pass through this set of points

- Check the second set of points with the first function

# Pint (-2 , 15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (-2)² - 2(-2) - 15 = 4 + 4 - 15 -7 ⇒ not same value of y-coordinate

∴ The graph of the function does not pass through point (-2 , 15)

∴ The graph does not pass through this set of points

- Check the third set of points with the first function

# Pint (-3 , 0)

∵ f(x) = x² + 2x - 15

∴ f(0) = (-3)² - 2(-3) - 15 = 9 + 6  -15 = 0 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-3 , 0)

# Pint (0 , -15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (0)² - 2(0) - 15 = -15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , -15)

# Pint (5 , 0)

∵ f(x) = x² + 2x - 15

∴ f(0) = (5)² - 2(5) - 15 = 25 - 10  -15 = 0 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (5 , 0)

∴ The graph passes through this set of points

* f(x) = x² - 2x - 15 passes through (-3 , 0) , (0 , -15) , (5 , 0)

- Check the first set of points with the second function

# Pint (0 , -15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(0)² - 2(0) + 15 = 15 ⇒ not same value of y-coordinate

∴ The graph of the function does not passes through point (0 , -15)

∴ The graph does not pass through this set of points

- Check the second set of points with the second function

# Pint (-2 , 15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(-2)² - 2(-2) + 15 = -4 + 4 + 15 = 15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-2 , 15)

# Pint (-1 , 16)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(-1)² - 2(-1) + 15 = -1 + 2 + 15 = 16 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-1 , 16)

# Pint (0 , 15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(0)² - 2(0) + 15 = 15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , 15)

∴ The graph passes through this set of points

* f(x) = -x² - 2x + 15 passes through (-2 , 15) , (-1 , 16) , (0 , 15)

- Now we have the first set of points and the third function

∴ The graph passes through this set of points

∴ f(x) = -x² + 2x - 15 passes through (0 , -15) , (1 , -14) , (2 , -15)

ANSWER

16

EXPLANATION

The given expression is;

[tex]3 {x}^{2} + 4 {y}^{2} [/tex]

If x=2, y=1 and z=-3, we substitute the values into the expression and solve.

We substitute to obtain;

[tex]3 {(2)}^{2} + 4 {(1)}^{2} [/tex]

We evaluate to get;

[tex]3 {(4)} + 4 {(1)}[/tex]

We multiply out to get:

[tex]12+ 4 = 16[/tex]

Therefore the value of the given expression is with the given values is 16

Answer:

11.77

Step-by-step explanation:

Volume of sphere = [tex]\frac{4}{3}[/tex] × π × r²

580 mm³ =  [tex]\frac{4}{3}[/tex] × π × r²

( Divide both sides by  [tex]\frac{4}{3}[/tex] )

435 mm³ = π × r²

( Divide both sides by π )

138.4648005 = r²

( Square root both sides )

11.76710672 = r

Answer: 5. 24.04

6. 30

7. 49.45

Step-by-step explanation:

use the law of sines for 5 and 6,

law of cosine for 7

5-6= opp/hyp.

7= adj/hyp

5. Let x be the missing angle.

We have the hypotenuse of the given right angle triangle to be 27 units.

The opposite side to the missing angle is 11 units.

We use the sine ratio to obtain:

[tex]\sin x=\frac{Opposite}{Hyppotenuse}[/tex]

[tex]\sin x=\frac{11}{27}[/tex]

[tex]x=\sin^{-1}(\frac{11}{27})[/tex]

[tex]x=24.04\degree[/tex] to the nearest hundredth.

6. Let y represent the missing angle.

We have the hypotenuse of the given right angle triangle to be 24 units.

The opposite side to the missing angle is 12 units.

We use the sine ratio to obtain:

[tex]\sin y=\frac{Opposite}{Hyppotenuse}[/tex]

[tex]\sin y=\frac{12}{24}[/tex]

[tex]y=\sin^{-1}(\frac{1}{2})[/tex]

[tex]y=30\degree[/tex].

7. Let the missing angle be z.

This time we have the adjacent side to be 13 units and the hypotenuse is 20 units.

We use the cosine ratio to obtain:

[tex]\cos z=\frac{Adjacent}{Hypotenuse}[/tex]

This implies that:

[tex]\cos z=\frac{13}{20}[/tex]

[tex]z=\cos ^{-1}(\frac{13}{20})[/tex]

[tex]z=49.46\degree[/tex] to the nearest hundredth

Answer:

C. a value close to the mean

Step-by-step explanation:

The positive skewness of distribution could be caused by a value close to the mean. Thus, option C is correct.

The normal distribution is somewhat similar where the main observation (mean or its surrounding) occurs frequently and as we go far from the mean, its chances decrease.

Normal distribution of proportion: The sampling distribution of the proportion we're talking about should be normally distributed.

A skewed distribution is a distribution having bias on one of the two sides (either left or right).

The positive skewness of distribution could be caused by a value close to the mean.

Thus, option C is correct.

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Answer:

138 cm.

Step-by-step explanation:

So first, we find the S.A. of the front and back.

The diagram says the side length of the front is 3 cm. and 3 cm.

3x3=9. So then, the back is also 9 cm, 9+9=18.

Now to find the S.A.'s of the four sides, you have to see the side lengths of each of them. The side lengths are 3 and 10.

3x10=30. This means each of them is 30 cm.

30x4=120. 120 is the total surface area of the four sides.

To find the total surface area of the whole rectangle, you add all the surface areas.

120+18=138 cm. (Not squared, since it's surface area and not area.)

Both terms [tex]a^8b^4[/tex] and [tex]a^2b^2[/tex] contain some powers of a and b. So, we can factor the occurrences with the smallest exponent:

[tex]a^8b^4+a^2b^2 = a^2b^2(a^6b^2+1)[/tex]

The complete factored form of the polynomial [tex]a^{8}b^{4} +a^{2}b^{2}[/tex]  is  [tex]a^{2}b^{2} (a^{6}b^{2} + 1 )[/tex] .

A complete factored form of expression is the result expression of the polynomial which is expressed as the product of its smallest factor format. We always get a simplified expression of the polynomial in the complete factored form.

The given expression is -  [tex]a^{8}b^{4} +a^{2}b^{2}[/tex]

Taking the term [tex]a^{2}b^{2}[/tex]  common to express the polynomial in factored form,

[tex]a^{8}b^{4} +a^{2}b^{2}[/tex]  =  [tex]a^{2}b^{2} (a^{6}b^{2} + 1 )[/tex]

Thus, the complete factored form of the polynomial [tex]a^{8}b^{4} +a^{2}b^{2}[/tex]  is  [tex]a^{2}b^{2} (a^{6}b^{2} + 1 )[/tex] .

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Answer:

-12

Step-by-step explanation:

think of it like this

Do -48/2 which equals -24 then cut that in half which equals -12 and thats your answer

Answer:

132.5 cm²

Step-by-step explanation:

A cuboid has 3 pairs of identical faces.

So the surface area is 2(L×W) + 2(L×H) + 2(W×H)

the calculation is 34.5 + 75 + 23 = 132.5 cm²

The total surface area of a cuboid with dimensions 7.5cm, 2.3cm, and 5cm is 132.5 cm², calculated using the formula for the surface area of a cuboid being 2lw + 2lh + 2wh.

The subject of this question concerns the calculation of the total surface area of a cuboid. A cuboid has six rectangular faces. To find the total surface area of the cuboid, we need to calculate the area of all six faces. The formula to find the surface area of a cuboid is 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.

In this case, we substitute the given dimensions into the formula to get: 2(7.5)(2.3) + 2(7.5)(5) + 2(2.3)(5) = 34.5 + 75 + 23 = 132.5 cm².

The total surface area of the cuboid is therefore 132.5 cm².

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The simplified expression for cos x csc x tan x is 1 .

Sure, let's simplify the expression step by step:

Given expression:[tex]\( \cos(x) \csc(x) \tan(x) \)[/tex]

We know that:

[tex]- \( \csc(x) = \frac{1}{\sin(x)} \)[/tex]

[tex]- \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)[/tex]

So, we substitute these into the expression:

[tex]\( \cos(x) \cdot \frac{1}{\sin(x)} \cdot \frac{\sin(x)}{\cos(x)} \)[/tex]

Now, we cancel out the common terms:

[tex]\( \frac{\cos(x) \cdot \sin(x)}{\sin(x) \cdot \cos(x)} \)[/tex]

Now, we can see that the numerator and the denominator cancel each other out:

[tex]\( \frac{1}{1} = \boxed{1} \)[/tex]

In conclusion, the simplified expression is ( 1 ).

We start by using the trigonometric identities to express [tex]\( \csc(x) \) and \( \tan(x) \) in terms of \( \sin(x) \) and \( \cos(x) \)[/tex]. Then, we substitute these expressions into the given expression. Next, we cancel out the common terms in the numerator and denominator, resulting in a simplified expression of 1. This simplification demonstrates the relationship between the trigonometric functions and highlights their interconnectedness through fundamental trigonometric identities.

Complete question:

Simplify the expression cos x csc x tan x

First add -2 and -6 together

Since these are both negitive signs you will add normally and and a negitive sign to the answer

-8

so you have...

-8 + 7

Since there is a negative (-8) and a positive (7) you will treat this as a normal subtraction problem, except your answer will have the sign of the biggest number

8 - 7 = 1

8 is the bigger number and has a negative sign therefore the answer is a negative number

so...

-8 + 7 = -1

-1

Hope this helped!

~Just a girl in love with Shawn Mendes

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Answer:

A. Perpendicular

Step-by-step explanation:

Lines are perpendicular if their slopes are opposite reciprocals of each other. Opposite, meaning if positive, the other slope is negative, and if negative, the other slope is positive. Reciprocal meaning, the number is flipped upside down, turning fractions into whole numbers and vice versa.

1/9  

-1/9

-9

The slopes are perpendicular

Answer:

This is a right triangle, so we know that:

h² = b' · c'

which is this case can be specificly written as:

BD² = AD · CD

BD² = 7 · 3 = 21

BD = √21

Now that we can also notice that ΔADB is also a right triangle, therefore we can apply the pythagorean theorem:

AD² + BD² = AB²

7² + (√21)² = x²

x²               = 49 + 21 = 70

x                 = √70

Answer:

C. Directrix and Focus

Step-by-step explanation:

Given choices are :

A. Locus and Directrix

B. Axis and vertex

C. Directrix and Focus or

D. Vertex and Locus

Now we need to find about which of the above choices are the exact same distance from a parabola.

By definition of parabola, vertex lies at equal distance from directrix and focus.

Hence choice  C. Directrix and Focus  is correct.

Answer:

C.Directix and Focus

Step-by-step explanation:

The directrix and the focus are both parts of the parabola that are the exact same distance form the vertex ot he parabola, the only difference is that they are in opposite directions, the focus of the parabola is always found inside of the parabola and in the axis of symmetry, on the same axis of symmetry both on the outside of the parabola, the same distance from the vertex than the focus you can find the directrix, which is a straight line that is perpendicular to the axis of symmetry.

Answer:  -5/21

Step-by-step explanation:

-3/7 & -4/6

common detonator is 42

-3/7 = -18/42 reduced to -9/21

-4/6 = -28/42 reduced to -14/21

difference between -9/21 and -14/21 = -5/21

Answer:

0.42% Chance Of The Counters Being Red

Step-by-step explanation:

1.00

-0.58

=0.42% Probability

The probability that a randomly picked counter from a bag containing only red and blue counters is red, given that the probability the counter is blue is 0.58, is 0.42.

The subject here is

probability

, which in

mathematics

is a measure of the likelihood that a particular event will occur. The problem states that the

probability

that a counter is blue is 0.58. Since we only have red and blue counters in the bag, and the probabilities of all possible outcomes must add up to 1, the

probability

that a counter picked at random is red is 1 - the

probability

that the counter is blue. So, to find the

probability

that the counter is red, subtract 0.58 from 1. The resulting

probability

that a randomly picked counter is red is therefore 0.42.

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Answer:

See attachment.

Step-by-step explanation:

The parent function is [tex]f(x)=|x|[/tex]

When this function is translated 2 units to the right, the new equation becomes; [tex]g(x)=|x-2|[/tex].

Another translation of 2 units up gives  [tex]h(x)=|x-2|+2[/tex].

A final dilation by a factor of [tex]\frac{1}{3}[/tex] gives  [tex]i(x)=\frac{1}{3}|x-2|+2[/tex].

The graph of this function is shown in the attachment.

Answer:

Its C

Step-by-step explanation:

On Edge

Answer:

[tex]y=-8.08[/tex] -------> [tex]y+8=3(x+2)^{2}[/tex]

[tex]y=14.25[/tex] -------> [tex]y-14=-(x-3)^{2}[/tex]

[tex]y=-7.625[/tex] -----> [tex]y+7.5=2(x+2.5)^{2}[/tex]

[tex]y=17.25[/tex] -------> [tex]y-17=-(x-3)^{2}[/tex]

[tex]y=-7.25[/tex] -------> [tex]y+7=(x-4)^{2}[/tex]

[tex]y=6.25[/tex] -------> [tex]y-6=-(x-1)^{2}[/tex]

Step-by-step explanation:  

we know that

The standard form of a vertical parabola is equal to

[tex](x-h)^{2}=4p(y- k)[/tex]

where

(h,k) is the vertex

the focus is (h, k + p)

and

the directrix is y = k - p

Part 1) we have

[tex]y+8=3(x+2)^{2}[/tex]

Convert to standard form

[tex](x+2)^{2}=(1/3)(y+8)[/tex]

The vertex is the point [tex](-2,-8)[/tex]

[tex]h=-2,k=-8[/tex]

[tex]4p=1/3[/tex]

[tex]p=1/12[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y=-8-(1/12)=-8.08[/tex]

Part 2) we have

[tex]y-14=-(x-3)^{2}[/tex]

Convert to standard form

[tex](x-3)^{2}=-(y-14)[/tex]

The vertex is the point [tex](3,14)[/tex]

[tex]h=3,k=14[/tex]

[tex]4p=-1[/tex]

[tex]p=-1/4[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y = 14-(-1/4)=14.25[/tex]

Part 3) we have

[tex]y+7.5=2(x+2.5)^{2}[/tex]

Convert to standard form

[tex](x+2.5)^{2}=(1/2)(y+7.5)[/tex]

The vertex is the point [tex](-2.5,-7.5)[/tex]

[tex]h=-2.5,k=-7.5[/tex]

[tex]4p=1/2[/tex]

[tex]p=1/8[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y=-7.5-(1/8)=-7.625[/tex]

Part 4) we have

[tex]y-17=-(x-3)^{2}[/tex]

Convert to standard form

[tex](x-3)^{2}=-(y-17)[/tex]

The vertex is the point [tex](3,17)[/tex]

[tex]h=3,k=17[/tex]

[tex]4p=-1[/tex]

[tex]p=-1/4[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y = 17-(-1/4)=17.25[/tex]

Part 5) we have

[tex]y+7=(x-4)^{2}[/tex]

Convert to standard form

[tex](x-4)^{2}=(y+7)[/tex]

The vertex is the point [tex](4,-7)[/tex]

[tex]h=4,k=-7[/tex]

[tex]4p=1[/tex]

[tex]p=1/4[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y=-7-(1/4)=-7.25[/tex]

Part 6) we have

[tex]y-6=-(x-1)^{2}[/tex]

Convert to standard form

[tex](x-1)^{2}=-(y-6)[/tex]

The vertex is the point [tex](1,6)[/tex]

[tex]h=1,k=6[/tex]

[tex]4p=-1[/tex]

[tex]p=-1/4[/tex]

the directrix is equal to

[tex]y = k-p[/tex] -----> [tex]y=6-(-1/4)=6.25[/tex]

The parabolas represented by y + 8 = 3(x+2)², y - 14 = -(x-3)², y - 17 = -(x-3)², and y - 6 = -(x-1)² match with the directrixes y = -7.25, y = 14.25, y = 17.25, and y = 6.25 respectively.

To match the equations representing parabolas with their directrixes, we need to use the fact that the equation of a parabola is given by y - k = a(x-h)², where (h,k) is the vertex of the parabola and the directrix is given by y = k - 1/4a.

Given this, we can match the equations as follows:
1. y + 8 = 3(x+2)² matches with y = -7.25
2. y - 14 = -(x-3)² matches with y = 14.25
3. y + 7.5 = 2(x+2.5)² there isn't a match in column B
4. y - 17 = -(x-3)² matches with y = 17.25
5. y + 7 = (x-4)² there isn't a match in column B
6. y - 6 = -(x-1)² matches with y = 6.25.

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The complete question here:

Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used

match the equations representing parabolas with their directrixes

Column A.

y+8=3(x+2)^2

y-14=-(x-3)^2

y+7.5=2(x+2.5)^2

y-17=-(x-3)^2

y+7=(x-4)^2

y-6=-(x-1)^2

Column B.

y=-7.25

y=6.25

y=17.25

y=14.25

Answer:

y = (x+ 3)^2 - 10.

Step-by-step explanation:

Vertex form is

y =  a(x - b)^2 + c

Here b = -3 and c = -10 so we have

y = a(x + 3)^2 - 10   where a is some constant.

The y intercept is (0, -1)  so substituting:

-1 = a * 3^2 - 10

-1 + 10 = 9a

9a = 9

a = 1

So the required parabola is (x+ 3)^2 - 10.

The answer to the rotation of triangle def is B)

Answer:

Step-by-step explanation:

To create triangle D'E'F', you need to rotate triangle DEF 180°, which results a figure with an opposite position, like a mirror. A 180° rotation always gives an opposite position, a mirror effect.

Basically just add them all together like normal addition except there is a z attached to each number:

43z + 15z + 7z + 5z + 46z + 14z

58z+ 7z + 5z + 46z + 14z

65z + 5z + 46z + 14z

70z + 46z + 14z

116z + 14z

130z

Hope this helped!

Answer:

130z

Step-by-step explanation:

43z + 15z + 7z + 5z + 46z + 14z

Since the all have z with a coefficient, they are all like terms

Factor out a z

(43 + 15 + 7 + 5 + 46 + 14 ​ )z

Then add all the coefficients together

(130)z

The total is 130z

Answer:

[tex]\dfrac{21}{32}=0.65625[/tex]

Step-by-step explanation:

If the probability of success is 50%, then p=0.5 and q=1-0.5=0.5.

At least three successes in six trials of a binomial experiment means that favorable are 3 successes, 4 successes, 5 successes and 6 successes.

1. 3 successes:

[tex]Pr_1=C^3_6p^3q^{6-3}=\dfrac{6!}{3!(6-3)!}\cdot (0.5)^3\cdot (0.5)^3=20\cdot \dfrac{1}{2^6}=\dfrac{5}{16}[/tex]

2. 4 successes:

[tex]Pr_2=C^4_6p^4q^{6-4}=\dfrac{6!}{4!(6-4)!}\cdot (0.5)^4\cdot (0.5)^2=15\cdot \dfrac{1}{2^6}=\dfrac{15}{64}[/tex]

3. 5 successes:

[tex]Pr_3=C^5_6p^5q^{6-5}=\dfrac{6!}{5!(6-5)!}\cdot (0.5)^5\cdot (0.5)^1=6\cdot \dfrac{1}{2^6}=\dfrac{3}{32}[/tex]

4. 6 successes:

[tex]Pr_4=C^6_6p^6q^{6-6}=\dfrac{6!}{6!(6-6)!}\cdot (0.5)^6\cdot (0.5)^1=1\cdot \dfrac{1}{2^6}=\dfrac{1}{64}[/tex]

Now, the probability of at least three successes in six trials of a binomial experiment is

[tex]Pr=Pr_1+Pr_2+Pr_3+Pr_4=\dfrac{5}{16}+\dfrac{15}{64}+\dfrac{3}{32}+\dfrac{1}{64}=\dfrac{20+15+6+1}{64}=\dfrac{42}{64}=\dfrac{21}{32}=0.65625[/tex]

To find the probability of at least three successes in six trials of a binomial experiment where the success rate is 50%, we'll need to consider the complement of this event, which is easier to calculate in this situation. The complement consists of the probability of either 0, 1, or 2 successes in the six trials. By finding the sum of these probabilities, we can subtract it from 1 to find the probability of the original event (3 or more successes).

First, let's recall the formula for the binomial distribution:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

where:
- P(X = k) is the probability of k successes in n trials,
- C(n, k) is the number of combinations of n items taken k at a time, it can be calculated using the formula C(n, k) = n! / (k! * (n - k)!),
- p is the probability of success for each trial,
- (1 - p) is the probability of failure for each trial,
- n is the number of trials, and
- k is the number of successes.

Since the success probability is 50%, or 0.5, and the complement includes the probability of 0, 1, or 2 successes, we can calculate each of these probabilities.

For k = 0 (zero successes):
P(X = 0) = C(6, 0) * (0.5)^0 * (0.5)^(6 - 0)
P(X = 0) = (6! / (0! * 6!)) * 1 * (0.5)^6
P(X = 0) = 1 * (0.5)^6
P(X = 0) = (1/64)

For k = 1 (one success):
P(X = 1) = C(6, 1) * (0.5)^1 * (0.5)^(6 - 1)
P(X = 1) = (6! / (1! * 5!)) * (0.5) * (0.5)^5
P(X = 1) = 6 * (0.5) * (0.5)^5
P(X = 1) = 6 * (1/64)

For k = 2 (two successes):
P(X = 2) = C(6, 2) * (0.5)^2 * (0.5)^(6 - 2)
P(X = 2) = (6! / (2! * 4!)) * (0.5)^2 * (0.5)^4
P(X = 2) = (15) * (0.25) * (0.0625)
P(X = 2) = 15 * (1/64)

Now we sum up these probabilities to get the complement:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
P(X < 3) = (1/64) + 6*(1/64) + 15*(1/64)
P(X < 3) = (1 + 6 + 15) / 64
P(X < 3) = 22 / 64
P(X < 3) = 11 / 32

Now to find the probability of at least three successes (P(X >= 3)), we subtract the complement from 1:
P(X ≥ 3) = 1 - P(X < 3)
P(X ≥ 3) = 1 - (11 / 32)
P(X ≥ 3) = (32 / 32) - (11 / 32)
P(X ≥ 3) = 21 / 32

Converting this to a percentage and rounding to the nearest tenth of a percent:
P(X ≥ 3) ≈ (21 / 32) * 100
P(X ≥ 3) ≈ 65.625%

Rounded to the nearest tenth of a percent, the probability is 65.6%.